Theorem rexv 2748

Description: An existential quantifier restricted to the universe is unrestricted. (Contributed by NM, 26-Mar-2004.)

Assertion

Ref	Expression
rexv	|- ( E. x e. _V ph <-> E. x ph )

Proof of Theorem rexv

Step	Hyp	Ref	Expression
1		df-rex 2454	. 2 |- ( E. x e. _V ph <-> E. x ( x e. _V /\ ph ) )
2		vex 2733	. . . 4 |- x e. _V
3	2	biantrur 301	. . 3 |- ( ph <-> ( x e. _V /\ ph ) )
4	3	exbii 1598	. 2 |- ( E. x ph <-> E. x ( x e. _V /\ ph ) )
5	1, 4	bitr4i 186	1 |- ( E. x e. _V ph <-> E. x ph )

Syntax hints:   /\ wa 103   <-> wb 104   E.wex 1485   e. wcel 2141   E.wrex 2449   _Vcvv 2730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-rex 2454  df-v 2732

This theorem is referenced by:  rexcom4  2753  spesbc  3040  abnex  4432  dfco2  5110  dfco2a  5111